Tom Waits stands before a backdrop of battered megaphones, not so much acknowledging the standing ovation as orchestrating it. And as yet he hasn't even played a note. He wiggles his fingers to denote that he requires the applause to become more vociferous, pretends to be buffeted by the ensuing roar when it does, removes his bowler hat and bows low.

Shamelessly milking a crowd who've endured some pretty byzantine anti-touting measures (you're required to turn up bearing not just your hugely expensive ticket but your passport and bank card), Waits has something of the fairground barker about him; which seems ironic, given his notoriously idiosyncratic attitude to self-promotion.

The last time he played Britain, Waits hadn't been here in 17 years, and it was for one show only - and was heralded by an interview in which he announced that he had no intention of coming here again for the foreseeable future.

The two Edinburgh dates on his Glitter And Doom tour arrive fewer than four years later - and by Waits' standards they must be positively hot on the heels - but they come in no less peculiar circumstances. He publicised them by posting a video on YouTube in which he claimed that the tour's schedule had been decided by astrology, and that those who cared to look might find an illuminating acronym in the first letters of the towns and cities he was visiting.

"PEHDTSCKJMBA," he reflected. "Pretty profound."

Like a character in one of his songs, Waits appears to be getting crankier and more abstruse with every passing year. Last night, there's a brief interlude where he takes to the piano and plays a selection of oldies: a heartbreaking Tom Traubert's Blues, and a version of Innocent When You Dream which gradually turns into a mass singalong. But shouted entreaties for more of the same fall on deaf ears. "Those are all requests," he frowns, "but they're your requests."

Back he goes to the newer stuff, for which the word leftfield might have been invented. Raddled Weimar oompah, careening Captain Beefheart-influenced blues, or a rough approximation of 60s Southern soul rendered deeply disturbing by Waits' mucoidal bark.

For a performer nearing 60 with a 35-year career behind him, that kind of approach is theoretically a recipe for disaster; but unlike many of his peers, Waits doesn't need to rely on back-catalogue nostalgia to provide a show's highlights.

Stamping his feet and causing billows of dust to rise from the stage, moving in exaggerated gestures, punctuating his songs with hilariously odd vaudevillian monologues, Waits is an utterly magnetic performer, capable of drawing you into the most difficult material.

He performs a version of Eyeball Kid which offers no tune whatsoever to cling on to, but still manages to be captivating rather than baffling.

As he declaims the lyrics, he dons a mirrored version of his bowler and turns slowly around, spotlights glinting off it like a discoball. It's eerie and transporting: cue another standing ovation.

At moments like that, it seems less like you've bought a ticket to a concert than to a different world. Suddenly, the fact you have to bring your passport along makes a strange kind of sense.